The-sum-of-first-10-terms-of-the-series-x-1-x-2-x-2-1-x-2-2-x-3-1-x-3-2-is-

Question Number 56424 by gunawan last updated on 16/Mar/19

The sum of first 10 terms of the series

{(x + \frac{1}{x})}^{2} + {(x^{2} + \frac{1}{x^{2}})}^{2} + {(x^{3} + \frac{1}{x^{3}})}^{2} + \dots is

Answered by mr W last updated on 16/Mar/19

a_{n} = {(x^{n} + \frac{1}{x^{n}})}^{2} = x^{2 n} + 2 + \frac{1}{x^{2 n}}

\underset{n = 1}{\sum^{10}} a_{n} = \underset{n = 1}{\sum^{10}} x^{2 n} + 10 \times 2 + \underset{n = 1}{\sum^{10}} \frac{1}{x^{2 n}}

= \frac{x^{2} (1 + x^{20})}{1 + x^{2}} + 20 + \frac{\frac{1}{x^{2}} (1 + \frac{1}{x^{20}})}{1 + \frac{1}{x^{2}}}

= 20 + \frac{{(1 + x^{20})}^{2}}{x^{18} (1 + x^{2})}

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